(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Biodiversity Heritage Library | Children's Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
Upload
See other formats

Full text of "Analytical Mechanics"

MOTION
81
PROBLEMS.
1.  Find the path and the velocity of a particle which moves so that
its position at any instant is given by the following pairs of equations:
(a)  x = at,           y = bt.
(b)   x = at,             y = at  | gt2.
(c)   x -at,              y ~ b cos coi.
(d)   x = a sin otf,    y = R
(e)   a; = a sin co;    ?/ = a cos oit.
(f)   a; = a sin at,   y = 6 sin co. / \     _    ^               _    _1.
\o/ *^    ^^ 7       y  ^^
2.  Prove the relation v = V-x2 + y'2 + z2.
81. Radial and Transverse Components of Velocity.  The magnitude of the velocity along the radius vector is, according to the results of the preceding section,
dr
-j7 dt
(D
The expression for the velocity at right angles to r is obtained by considering the motion of the projection of the particle along a perpendicular y to r. When the particle moves through cfe, its projection moves through r dd, Fig. 50, therefore the required velocity
is
rdd '' dt
,r-rt.
dt
(2)
FIG. 50.
The components vr and vp may be expressed in terms of x and y by differentiating the equations of transformation
r2=x2+2/2                                    (3)
and
(4)